analysis of an industrial continuous slurry reactor for ethylene–butene copolymerization

Analysis of an industrial continuous slurry reactor for

ethylene–butene copolymerization*

Cristiano H. Fontesa,*, Mario J Mendesb,1

aDepartamento de Engenharia Quımica, Escola Politecnica, Universidade Federal da Bahia Rua Aristides Novis, 2, Federacao, Salvador-Ba, BrasilbDepartamento de Engenharia de Sistemas Quımicos, Faculdade de Engenharia Quımica, Universidade Estadual de Campinas,

Cidade Universitaria Zeferino Vaz,.Campinas-SP, Brasil

Received 12 November 2004; received in revised form 23 December 2004; accepted 25 January 2005

Available online 10 March 2005

Abstract

This work presents the analysis of a slurry polymerization stirred tank reactor for the production of high-density polyethylene. A

coordination mechanism is adopted including initiation, propagation, first order deactivation, hydrogen transfer, ethylene transfer, transfer to

co-catalyst and b-hydride elimination. Two site types are considered each one with its own kinetic constants. A non-uniform solid phase is

considered and in the particle there is a radial distribution of chemical species such as the living and dead polymer chains. In this sense, a

general local balance is proposed, providing state equations for the moments of chain length distribution and for the active site concentrations

that are treated together with all others modeling scales. The multigrain model approach was adopted, combined with a simulation strategy

applying orthogonal collocation. The simulation procedure handles all the equations simultaneously and the model is capable of predicting

the behavior of operation variables and variables associated with the polymer properties.

q 2005 Elsevier Ltd. All rights reserved.

Keywords: Olefin polymerization; Multigrain model; General local balance

1. Introduction

Continuous slurry polymerization with heterogeneous

Ziegler–Natta catalysts is one of the most employed

processes for the production of polyolefins. On the other

hand, continuous polymerization processes are good

candidates for MPC (Model Predictive Control) because

they are multivariable, have some unusual dynamics and

have more manipulated variables than controlled ones

(Schnelle and Rollins [1], Moudgalya and Jaguste [24], Qin

and Badgwell [25]). MPC optimizes, over the manipulated

0032-3861/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.polymer.2005.01.091

Abbreviations: NAMW, number-average molecular weight; WAMW,

weight-average molecular weight; ODE, Ordinary differential equation.

* This paper was presented in a simplified form at the II Congresso de

Engenharia de Processos do Mercosul (ENPROMER’99) August 30–

September 02 1999, Florianopolis, Brazil.* Corresponding author. Tel.: 55 71 203 9801; fax: C55 71 203 9802.

E-mail addresses: [email protected] (C.H. Fontes),

[email protected] (M.J. Mendes).1 Tel.: 13081 70, (55) (19) 3788-3941, fax: (55) (19) 3788 3910.

inputs, forecasts of process behavior. The forecasting is

accomplished with a process model, and therefore, the

model is the essential element of a MPC controller

(Rawlings [2]). The objective of this work is the develop-

ment of a phenomenological dynamic model of an

industrial continuous slurry tank reactor for the production

of high-density polyethylene through copolymerization of

ethylene-1-butene with a Ziegler–Natta catalyst. Such

phenomenological models, based on first principles, are

naturally of a prohibitive dimension to be used in the frame

of a NMPC controller for the reactor. However, aside from

their eventual academic value, they can find other important

uses, like that of training plant operating staff, or playing the

role of the ‘process’ in evaluating the applicability of

current advanced control tools (Schnelle and Rollins [1],

Ozkan et al., [26]).

The analysis of slurry polymerization reactors is a

relatively well-explored field (Ray [3], Kiparissides [4],

Dube et al., [5]). In solid-catalyzed polymerizations,

polymer grows at the active sites on the catalyst until

chain transfer occurs, forming ‘dead’ polymer chains. In

most cases the particle remains intact so that a single

polymer particle is grown, or ‘replicated’, from each

Polymer 46 (2005) 2922–2932

www.elsevier.com/locate/polymer

http://www.elsevier.com/locate/polymer

Nomenclature

ap specific surface area of the macroparticle

(cm2/g)

A, B matrices for account the derivatives of the node

orthogonal polynomial at the interpolation

points

ci concentration of species i in the macroparticle,

mol/L

csi concentration of species i in the microparticle,

mol/L

cik concentration of specie i in the k collocation

point, mol/L

Cd deactivated site

D(n, m) dead polymer chains with n monomer units and

m comonomer units, and their concentration,

mol/L

Fi feed flow rate of component i, kg/h

Ki mass transfer coefficient of component i, cm/min

kp rate of propagation constants, L/(mol$min)

kt,s rate constant for b-hydride elimination, minK1

kt other rate constants for chain transfer,

1/(mol$min)

kd rate constant for deactivation, minK1

mgi mass of component i in the reactor headspace,

ton

mli mass of component i in the reactor liquid phase,

ton

mst mass of component i in the reactor solid phase,

ton

mti total mass of component i in the reactor, ton

mp total mass of polymer inside the reactor, ton

ngi moles of component i in reactor gas had-space

NCL number of internal collocation points

P1(n, m) living polymer chains with n monomer units and

m comonomer units, with terminal monomer,

and their concentration, mol/(Lcatal)

P2(n, m) living polymer chains with n monomer units and

m comonomer units, with terminal comonomer,

and their concentration, mol/(Lcatal)

Pr reactor pressure, kgf/cm2

P(0,0), P0,0 concentration of active sites, mol/(Lcatal)

PsNX vapor pressure of solvent, kgf/cm2

q volumetric flow of slurry, m3/h

r radial position in growing polymeric particle,

cm

rs radial position in the growing microparticle, cm

R radius of macroparticle, cm

R0 initial radius of macroparticle, cm

Rc radius of catalyst fragment, cm

Rs radius of microparticle, cm

Ri rate of production of species i per unit volume of

catalyst, mol/(Lcatal min)

Rvi rate of production of species i per unit volume of

macroparticle, mol/(Lcatal min)�Ri mean rate of production of species i per unit

mass of polymer in macroparticle, mol gK1

minK1

Rpol production rate, ton/min

Rpjð0;0Þrate of active sites of type j

Tr reactor temperature, 8C

Te temperature of feed stream, 8C;

t time

Vg volume of gas head-space in reactor, m3

Vl volume of liquid phase in slurry, m3

Vs volume of solid phase in slurry, m3

Vsl volume of slurry inside the reactor, m3

Vc catalyst volume inside reactor, L_Vcs volumetric flow of catalyst withdrawal, L/min_Vce volumetric flow of catalyst feed stream, L/min

w1 monomer molecular weight

w2 comonomer molecular weight

x mass fraction of component i in the liquid phase

y mass fraction of component i in the gas head-

space

DHR,pol heat of polymerization, kJ/kg

h sorption factor

r density

Subscripts:

p polymer

i specie or component

j site type

Superscripts:

o catalyst feed stream

j site type

C.H. Fontes, M.J. Mendes / Polymer 46 (2005) 2922–2932 2923

catalyst particle. Commercially produced polyolefins have

polydispersities ranging from 3 to 20, while from theory for

standard olefin polymerization kinetics an ultimate poly-

dispersity of 2 is predicted. The broad MWD (Molecular

Weight Distribution) in olefin polymers is attributed, on one

hand, to monomer diffusion resistance in the growing

polymer particle; on the other hand, this broad MWD has

also been linked to catalytic site heterogeneity (Soares and

Hamielec [6], Floyd et al. [7], Mckenna and Soares [8]. In

this way, an efficient model of a slurry polymerization

reactor should include the transport in the polymer phase

and the heterogeneity of the catalytic sites.

Fig. 1 represents schematically the reactor. Ethylene,

1-butene (comonomer) and hydrogen (chain transfer agent),

together with n-hexane (solvent), Ziegler– Natta type

catalyst and co-catalyst are fed continuously to the reactor,

while the solid–liquid slurry, present in the reactor together

with a gas phase, is continuously removed from the reactor.

Fig. 1. Schematic representation of the reactor.

C.H. Fontes, M.J. Mendes / Polymer 46 (2005) 2922–29322924

For the control of the reactor temperature, heat is exchanged

in the reactor jacket and by circulation of the solid–liquid

slurry through external heat exchangers.

The analysis of a simple chemical reactor considers

normally three main hierarchical levels (Aris [9]). These

levels are naturally found in the analysis of a slurry

polymerization reactor, as shown in Table 1.

The understanding and description of the phenomena at

each one of these levels and of the connections between the

levels leads to the model of the reactor.

2. Kinetics of heterogeneous Ziegler–Natta

polymerization

Ziegler–Natta catalysts are anionic coordination cata-

lysts (Kiparissides [4]) (Table 2).

For heterogeneous Ziegler–Natta catalysis, even on the

same catalyst particle, different active sites can have

different propagation rate constants, which as seen above

can give rise to very broad molecular weight distributions.

Two types of catalytic sites are considered in this work

(Table 2). Implicit in the kinetic equations is the concept of

the terminal model for copolymerization (Soares and

Hamielec [6]). The absence of a site activation step in the

proposed mechanism is explained by the assumption of the

existence of a prepolymerization operational stage before

the reactor feeding. This assumption is related to the

cocatalyst effect on the dynamic behavior of reactor.

According to the mechanism of Table 2 and using the

global constants

kt;ghj Z kt;sj Ckt;hjCsH2

; (1a)

Table 1

Hierarchical scales of the analysis of a slurry polymerization reactor (Ray [3])

1. Microscale: Polymerization kinetics; Nature of active sites;

2. Mesoscale: Inter- and intraphase heat and mass transfer;

3. Macroscale: Overall material and energy balances; Macromixing.

kt;ccejZ kt;ccj

CsCC Ckt;mejCsET; (1b)

kt;ccbjZ kt;ccj

CsCC Ckt;mbjCsET; (1c)

the rate of production of the living chains P1j (n,m) per unit

volume of catalyst is:

RPljðn;mÞZK kt;hgj Ckp;eejCsET Ckp;ebjCsBT Ckt;ccej

�

Ckd;j

�P1jðn;mÞCkp;eejCsETP1jðnK1;mÞ

Ckp;bejCsETP2jðnK1;mÞ ðR1;mR1Þ (2)

For well-known reasons (Ray [10], Hutchinson et al. [11]),

the state of the polymer was represented mathematically

using the method of moments, so that the model obtained

provides only the prediction of average polymer properties.

The (ik) th moments of living (growing) chains ended with

monomer or comonomer at site type j are

lik1j ZXNnZ1

XNmZ0

nimkP1jðn;mÞ (3a)

and

lik2j ZXNnZ0

XNmZ1

nimkP2jðn;mÞ (3b)

respectively, while

Lik ZXNnZ0

XNmZ0

nimkDðn;mÞ (3c)

is the (ik)th moment of the dead chains. Rate equations for

the zero, first and second order moments for each active site

type ðl001j; l00

2j;L00

j ; l101j; l10

2j;L10

j ; l011j; l01

2j;L01

j ; l111j; l11

2j;L11

j ;

l201j; l20

2j;L20

j ; l021j; l02

2jand L02

j are obtained from the balances

of living and dead chains and from definitions (Eqs. (3a–c)).

Thus, for example, the rate of production per unit volume of

catalyst of the zero order moment l001j

is given by (Fontes

[12]):

Rl001jZKðkd;j Ckp;ebjCsBT Ckt;hgiÞl

001j

Ckp;eejCsETPjð0; 0ÞCkp;bejCsETl002j Ckt;ccbjl

002j

(4)

For each site type there are thus 18 rate equations for the

moments of zero, first and second order. On the other hand,

the production rates per unit volume of catalyst of ethylene,

1-butene and hydrogen, at each type of active site j, are

given by:

Table 2

The kinetic mechanism of copolymerization

Site type j(jZ1,2)

KInitiation :

Pjð0; 0ÞCET��/kp;eej P1jð1; 0Þ

Pjð0; 0ÞCBT��/kp;bbj P2jð0; 1Þ

KPropagation :

P1ðn;mÞCET��/kp;eej P1jðnC1;mÞ

P2ðn;mÞCET��/kp;bej P1jðnC1;mÞ

P1ðn;mÞCBT��/kp;ebj P2jðn;mC1Þ

P2ðn;mÞCBT��/kp;bbj P2jðn;mC1Þ

8>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>:

�Chain Transfer :

P1jðn;mÞ ��/kt;sj Pjð0; 0ÞCDjðn;mÞ

P2jðn;mÞ ��/kt;sj Pjð0; 0ÞCDjðn;mÞ

P1jðn;mÞCH2 ��/

kt;hj Pjð0; 0ÞCDjðn;mÞ

P2jðn;mÞCH2 ��/

kt;sj Pjð0; 0ÞCDjðn;mÞ

P1jðn;mÞCET��/kt;mej P1ð1; 0ÞCDjðn;mÞ

P2jðn;mÞCET��/kt;mbj P1ð1; 0ÞCDjðn;mÞ

P1jðn;mÞCCC��/kt;ccj P1ð1; 0ÞCDjðn;mÞ

P2jðn;mÞCCC��/kccj P1jð1; 0ÞCDjðn;mÞ

�Deactivation :

P1jðn;mÞ ��/kd;j CdjCDjðn;mÞ

P2jðn;mÞ ��/kd;j CdjCDjðn;mÞ

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

Fig. 2. Multigrain approach for the particle growth.


RETj ZK½kp;eejðl001j CPjð0; 0ÞÞCkt;mejl

001j Ckp;bejl

002j

Ckt;mbjl002j �CsET; (5)

RBTj ZK kp;bbjðl002j CPjð0; 0ÞÞCkp;ebjl

001j

� �CsBT; (6)

RH2jZK kt;hjðl

002j Cl00

1jÞ

j kCsH2; (7)

while the rate of production of each active site is:

RPjð0;0ÞZKðkp;eejCsET Ckp;bbjCsBT Ckd;jÞPjð0; 0Þ

Ckt;hgjðl001j Cl

002j Þ: (8)

Considering the existence of two types of monomer units

and of two site types, one obtains finally for the number-

average molecular weight (NAMWZMn), for the weight-

average molecular weight (WAMWZMw), and for the

fraction of comonomer incorporated into the polymer (XBT):

Mn Zw1

P2jZ1 l

10jP2

jZ1 l00j

Cw2

P2jZ1 l

01jP2

jZ1 l00j

; (9)

Mw Z

P2jZ1ðw

21l

20j C2w1w2l

11j Cw2

2l02j ÞP2

jZ1ðw1l10j Cw2l

01j Þ

; (10)

XBT Z

P2jZ1 l

01jP2

jZ1ðl01j Cl10

j Þ(11)

with likj Zlik1jClik2jCLikj . The polydispersity index Ip is

calculated directly from Mw and Mn (Hutchinson [11])

Ip ZMw

Mn

: (12)

Thus, not only the rates of production of the chemical

components but also the important mean properties of the

copolymer can be expressed as functions of the moments of

zero, first and second order.

3. Transport with reaction in the polymer particles

As mentioned above, it is generally assumed that both

phenomena, namely multiplicity of active site types and

mass and heat transfer within the polymer particle, might

play a role in the broadening of the MWD of polymers

produced with supported Ziegler–Natta catalysts. From the

several types of models that include inter- and intraparticle

transport (Dube et al. [5], Mckenna and Soares [8]), the

multigrain model (Mckenna and Soares [8], Sun et al. [21]),

schematically represented in Fig. 2, was used in the present

work.

This model considers the complete rupture of the catalyst

at time zero, leading to the existence of two different levels

for transport, the macro and microparticle levels. It will be

assumed that the intraparticle temperature gradients are


negligible (Mckenna and Soares [8]). Neglecting the

convective mass transfer inside the macroparticle and the

instantaneous variation of its volume, a global material

balance for component i (iZET, BT, H2) on the macro-

particle leads to the equations

vcivt

Z2

rDeff;i

vcivr

CDeff;i

v2ci

vr2KRvi: (13)

Eq. (13) are solved with the following initial and boundary

conditions:

r Z 0 :vcivr

Z 0; (13a)

r ZRðtÞ : Deff;i

vcivr

ZKi½cli KciðR; tÞ�; (13b)

tZ 0 : ciðr; 0ÞZ ci0: (13c)

Deff,i is the effective diffusivity of species i in the

macroparticle. Rvi, the rate of production of species i per

unit volume of the macroparticle, is related with Ri, the total

rate of production of i per unit volume of catalyst (Eqs. (5)–

(7)).

Ri ZX2

jZ1

Rij (14)

and

Rvi Zð1K3ÞRi

f3: (15)

3 is the macroparticle porosity, which will be assumed to be

uniform and constant in the macroparticle, and f is the

microparticle growth factor (Eq. 18). Monomer, which

diffuses through the macroparticle pores, adsorbs on the

layer of polymer surrounding the catalyst fragments, and

diffuses through this layer to the active sites on the surface

of those fragments.

The mass balance equations for the microparticle are

then:

vcsi

vtZDsi

1

r2s

v

vrs

r2s

vcsi

vrs

� �; (16)

with

rs ZRsðtÞ : csi Z hici; (16a)

and

rs ZRc : 4pR2cDsi

vcsi

vrs

ZK4

3pR3

cRi: (16b)

hi is the sorption factor (equilibrium at the solid–liquid

interface). A uniform distribution of cocatalyst is supposed

to exist on the solid phase. Employing the Quasi Steady

State Approximation for the material balance of the

microparticle (Hutchinson [11]), one obtains for the con-

centration of each component at the surface of the catalyst

fragment:

c0si Z hici K

R2c

3Dsi

Ri 1K1

f

� �; (17)

where the microparticle growth factor f is defined by

fðr; tÞZRsðr; tÞ

Rc

(18)

f3(r,t) is thus the ratio of the total volume of the

microparticle to the volume of the catalyst fragment in the

microparticle at radial position r and instant t, and its value

can be obtained from the moments of the chain length

distribution (Fontes [12]):

f3 Z 1Cw1

P2jZ1 l

10j Cw2

P2jZ1 l

01j

rp

(19)

The assumption of a non-uniform species distribution on the

macroparticle, intrinsic to the multigrain model, leads to a

radial variation of the moments and of the active site

concentrations. These distributions are given by the

equations (see Appendix):

vlik1

vtZRlik

1K

_Vce

Vc

lik1 ; (20)

vlik2

vtZRlik2

K_Vce

Vc

lik2 ; (21)

vLik2

vtZRlik

2K

_Vce

Vc

Lik2 ; (22)

vP0;0

vtZRP0;0

C_Vce

Vc

ðPo0;0 KP0;0Þ; (23)

where

lik1 ðr; tÞZX2

jZ1

lik1jðr; tÞ; lik2 ðr; tÞZ

X2

jZ1

lik2jðr; tÞ;Likðr; tÞ

ZX2

jZ1

Likj ðr; tÞ:

The newly formed polymer chains push the previously

formed polymer layer, thus increasing the radius of the

microparticles and consequently the size of the macro-

particles. Let dVm be the volume of the macroparticle

between r and rCdr and dVsd the volume of the solid

(polymerCcatalyst), that is, the volume of the microparti-

cles, in dVm. Then

dVm ZdVsd

ð1K3ÞZ 2pR3g1=2dg; (24)

where g1/2Zr/R. With the definition of the growth factor f

the volume of catalyst in dVm is then given by


dVcatðg; tÞZdVsd

f3Z 2pR3 ð1K3Þ

f3g1=2dg;

or

Vcat Z4

3ð1K3ÞpR3

0 Z 2pð1K3ÞR3

ð1

0ð1=f3Þg1=2dg: (25)

Eq. (25) can be written in the form

F3m Z

R

R0

� �3

Z2

3Ð 1

0 ð1=fÞ3g1=2dg

; (26)

where Fm is a ‘global macroparticle growth factor’.

For the mean rate of production of component i per unit

mass of polymer, �Ri, one obtains

�Ri ðtÞZ

ÐVm

RvidVmÐVm½w1ðl

101 Cl10

2 L10ÞCw2ðl011 Cl01

2 L01Þ�½ð1K3Þf3�dVm

;

(27)

while the specific surface area of the macroparticle is

apðtÞZ

4pR2ÐVm½w1ðl

101 Cl10

2 L10ÞCw2ðl011 Cl01

2 L01Þ�½ð1K3Þ=f3�dVm

:

(28)

4. Macroscale modeling

Based on the operational practice and on information

available in literature some simplifying assumptions were

adopted:

(1)
The reactor level (total slurry volume) is held constant.
(2)
The temperature in the reactor is uniform (Floyd et al.
[13], Soares and Hamielec [14]).

(3)
Thermodynamic equilibrium exists at the gas–liquid
interface (Floyd et al. [7], Mckenna and Soares [8]).

(4)
The gas and slurry phases are well mixed.
It should be considered that the well-mixed character-

istics of the phases in reactor generate a residence time

distribution for the macroparticles in reactor. To capture the

effects of such a distribution, population balance techniques

should be used in the construction of the model of the

reactor (Zacca et al. [15]). There is, however, some evidence

that for sufficiently active catalysts the effect of particle size

distribution might be of less importance (Galvan and Tirrel

[16]).

With the above hypotheses the mass balances for

component i in the liquid, gas and solid phases take the

form:

dðmgi Cml

iÞ

dtZFi K

q

Vs CVl

mli KKiapðc

li KciðR; tÞÞ; (29)

dmsi

dtZK�Rimp K

q

Vs CVl

msi KKiapmpðc

li KciðR; tÞ: (30)

mgi ;m

li and ms

i are the masses of component i present in the

gas, liquid and solid phases of the reactor, VslZVsCVl is the

slurry volume in reactor, q is the volumetric flow rate of

slurry leaving the reactor and �Ri is the mean rate of

production of i per unit mass of the solid present in the

reactor.

Overall mass and energy balances are:

dP

i mgi C

Pi n

li C

Pi m

si

� �dt

ZXi

Fi Kq

Pi m

li C

Pi m

si

� �Vl CVs

; (31)

Xi

mtiCpi

dTrdt

ZKXi

FiCpiðTr KTeiÞK _Qc K _Qt

C ðKDHR;polÞRpol; (32)

where Rpol is the global rate production of polymer, and _Qc

and _Qt are the heat transfer rates in the reactor jacket and in

the external heat exchangers, respectively.

5. Simulation

No effort has been made to identify the model

parameters; typical literature values were used instead.

The constant volume restriction

Vs CVl ZXi

mli

riC

Xi

msi

riZVsl; (33)

and the hypothesis of ideal gas behavior (Soares and

Hamielec [14]),

Pr ZXi

ngi

RgTr

Vg

; (34)

provided additional algebraic equations of the model.

Henry’s law was used to calculate the concentration of

monomer, comonomer, hydrogen and nitrogen in the liquid

phase,

yiPr Z hixi; ðiZET;BT;H2;N2Þ; (35)

while Raoult’s law was applied to the solvent (Soares and

Hamielec [14]).

yNXPr Z xNXPsNXðTrÞ (36)

Table 4

Diffusivities (cm2 minK1)

Species Macroparticle Deff Microparticle Dsi

Ethylene 1.2!10K4 6!10K5

1-butene 1!10K4 6!10K5

Hydrogen 3.6!10K4 6!10K5

Nitrogen 0.6!10K4 –

n-hexane 1.5!10K4 –


The Henry constants and densities as functions of

temperature were obtained using available correlations

(Freitas [17]). The correlation of Marrone and Kirwan

[18] was used to calculate the liquid–solid mass transfer

coefficients. Table 3 shows the values of the frequency

factors for the rate constants (Freitas [17]).

For the activation energies of the propagation, chain

transfer and deactivation steps, respectively, the values of

29.4 kcal/mol, 50.2 kJ/mol and 4.2 kJ/mol were taken.

Typical values for the macroparticle and microparticle

diffusivities are shown in Table 4 (McAuley et al. [19],

Mckenna et al., [20]). The macroparticle porosity 3 was

assumed constant and equal to 0.4 (McAuley et al. [19]).

The sorption factor hi (solid–liquid interface) was assumed

to be unity.

With the exception of the equations of the multigrain

model, the phenomenological model presented above is

made of ordinary differential and algebraic equations. The

equations of the multigrain model were integrated using the

classical orthogonal collocation method (Villadsen and

Michelsen [22]). Following this approach, Eq. (13) are

transformed into:

dci;k

dtZ

6

R2Deff;i

XNCLC1

jZ1

Akjci;j

C4gk

R2Deff;i

XNCLC1

jZ1

Bkjci;j KRi;k; ðkZ 1;.NCLÞ;

(37)

with the boundary condition:

gZ 1 :2Deff;i

R

XNCLC1

jZ1

ANCLC1;jcij

ZKi cli Kci;NCLC1

� �: (38)

NCL is the number of interior collocation points. Gaussian

Quadrature was used to integrate equation (26) for the

radius of the macroparticle.

Table 3

Frequency factors of the kinetic constants (L.molK1.min-1)

Site 1 Site 2

Propagation:

Kp,ee 1.101!108 1.101!108

Kp,be 2.591!106 1.943!107

Kp,eb 8.290!107 8.290!107

Kp,bb 1.943!106 8.031!106

Chain transfer

Kt,s (minK1) 1.615!105 1.615!105

Kt,h 1.421!108 1.421!107

Kt,me 3.392!106 3.392!105

Kt,mb 3.392!106 1.615!105

Kt,cc 3.876!107 3.876!106

Deactivation

Kd (minK1) 3!10K1 3!10K1

With two types of catalytic sites 36!(NCLC1)C2!(NCLC1) ordinary differential equations were thus

obtained for the multigrain model, comprising the balances

for the moments of the chain length distribution and the

active sites. Table 5 incorporates all the equations of the

complete dynamic model. Thus, for example, using five

interior collocation points, NCLZ5, the differential alge-

braic equation system of the model comprises 14 algebraic

equations and 267 ordinary differential equations.

A differential-algebraic system equation solver (Le

Roux [23]) was employed for the simulation. Two sets of

simulation tests were performed. A first set of tests was

intended to evaluate the model through the response to

different step changes of the feed rates of hydrogen,

catalyst, comonomer, solvent (n-hexane) and co-catalyst.

These inputs were selected due to their expected effects

on the process outputs. Table 6 presents the magnitude of

each input step change and the time of its application.

With the exception of 1-butene (comonomer), all the

imposed changes represented feasible operational

changes. In normal operation conditions the feed rate of

the comonomer is much lesser than that of ethylene, so

that a high step change FBT was imposed in order to

enable the detection of its effect on the process variables

and on the polymer properties.

In all simulation tests, the initial steady state was

obtained using the dynamic model itself (false transient);

the initial state obtained in this way was used as the basis for

the data normalization. Table 7 presents the absolute initial

values of all input variables together with initial values for

the weight-average molecular weight (WAMW), number

average molecular weight (NAMW) (output variables), and

temperature (state variables), according to Figs. 3–5.

Fig. 3 presents the influence of the number of types of

catalytic sites on the variation of the polydispersity due to

the step input changes defined in Table 6. In both cases all

the other model parameters were the same. The results of

Fig. 3b showed a polydispersity index close to 2 and not too

sensible to the perturbations imposed on the feed rates,

while for Fig. 3a the polydispersity index was always

greater than 2 and showed sensible variations with the input

changes. These results show that, under the polymerization

conditions assumed in this work, the effect of multiple types

of catalytic sites on the polymer properties (polydispersity

index) is more important than that of mass transfer

resistances in the macroparticle.

Table 5

Dimension of the complete dynamic model using orthogonal collocation for the multigrain equations.

Number of equations

Statistic moments 18!2!(NCLC1)

Active sites balances 2!(NCLC1)

Macroparticle mass balances 5!NCL

Algebraic equations 14

Component mass balances 12

Overall energy and mass balances 2

Total of equations 18!2!(NCLC1)C2!(NCLC1)C5!NCLC28


Fig. 4 presents the results for the changes of the weight-

average molecular weight (WAMW) and of the number

average molecular weight (NAMW) due to the input

changes. The increase in hydrogen concentration led

immediately to a decrease of WAMW and of NAMW,

due to an intensification of hydrogen chain transfer. The

model also predicted an increase in the polydispersity index

due to a raise in hydrogen concentration, as expected

(Fontes [12]). The effect of the comonomer feed rate on

the polidispersity can be assigned to the increase in the

comonomer composition and to the deviation from the homo-

polymerization conditions. The small effect of the catalyst

feed rate on the polydispersity index can be attributed to the

simultaneous reduction in both WAMW and NAMW. The

reduction in these average molecular weights due to

increase in the catalyst feed rate is assigned to the raise in

active site concentration. The n-hexane effect in the polymer

properties is due to increase in temperature.

Although the ratio of propagation rate to transfer rate for

1-butene was assumed to be smaller than that for ethylene,

the increase in the WAMW, due to the positive step in the

Table 6

Step changes on the main input variables (simulation test).

Step input Application instant, (min)

Hydrogen feed flow (FH2) (C50%) 25

Catalyst feed flow(FC) (C33%) 500

Comonomer feed flow (FBT) (C1000%) 1000

Solvent feed flow (FNX) (-19%) 1500

Cocatalyst feed flow (FCC) (C250%) 2000

Table 7

Initial absolute values for simulation tests (figures 3–5, two active sites)

Hydrogen feed flow 10 kg/h

Catalyst feed flow 0.075 kg/h

Comonomer feed flow 150 kg/h

Solvent feed flow 16000 kg/h

Cocatalyst feed flow 0.2 kg/h

Monomer feed flow 7500 kg/h

Water flow in the reactor jacket 150000 kg/h

Water flow in the external heat external heat

exchangers

60000 kg/h

WAMW 122000

NAMW 29017

Temperature 84.6 8C

comonomer feed (tZ1000 min), is attributed to the higher

molecular weight of the comonomer. In this case, an

opposite effect was observed on the NAMW.

Fig. 5 shows the temperature behavior. The strong effect

of the comonomer feed rate is due to the high step change

imposed in this case.

Figs. 3–5 show that the response of process variables

such as temperature is typically more rapid than that of the

polymer properties (WAMW, NAMW, polydispersity

index), confirming the high value of the time constants

associated with the reactional variables. On the other hand, a

relatively low effect of the cocatalyst feed rate on all output

variables analyzed was observed. The inclusion of a site

activation step in the kinetic model (Table 2) would have

increased this effect.

Fig. 3. Influence of the number of types of catalytic sites on the reactor

dynamics (polydispersity index): (a) kinetic model with two types of

catalytic sites; (b) kinetic model with one type of catalytic site.

Fig. 4. Dynamic response of: (a) normalized WAMW; (b) normalized

NAMW.

Fig. 6. Normalized hydrogen/ethylene ratio for two distinct operational

runs.


A second set of simulation tests was accomplished in

order to evaluate the predictive character of the model,

using values of the hydrogen/ethylene ratio in the reactor

gas phase measured by chromatographic analysis for three

distinct operational runs (Figs. 6 and 7). These data, together

with the input flows, are reliably and were collected directly

from the data acquisition system. Despite the noisy behavior

of the plant data, the model predicts in both cases the

dominant tendency of the process dynamics. Furthermore,

the quality of the results shown in Figs. 6 and 7 is good

indeed, considering that the data collected from the plant

included no values of the hydrogen flow rate.

Fig. 5. Variation of the normalized reactor temperature due to input step

changes.

The high fluctuations in the plant data are due to

perturbations occurring in the feed streams (catalyst,

hydrogen, and ethylene). Fig. 7 shows also that the ethylene

consumption increases rapidly with an increase of the

catalyst feed rate, leading to a pronounced increase of the

hydrogen/ethylene ratio in the reactor gas phase.

The advantages of the orthogonal collocation technique

as a tool to solve the equations of the multigrain model

should be adequately stressed. Table 8 compares the

computational time for the simulation of 1500 min of

reactor operation time using two different types of

numerical procedures, the finite difference method with 25

shells (Hutchinson [11]) and the orthogonal collocation

technique with 5 interior points (the model included only

one type of site). As it is observed, the dimension of the

numerical models, and consequently the computational

Fig. 7. Normalized hydrogen/ethylene ratio for an operational run.

Table 8

Solution of the dynamic model.

Discretization method for

multigrain equations

Number of

ODE’s Simulation time, min

Finite difference (25

shells)

608 134

Orthogonal collocation

(5interior points)

153 9


times, were quite different, with an expressive advantage for

the use of the orthogonal collocation technique.

6. Conclusions

The complex nature of slurry polymerization systems,

related to the existence of different types of mechanisms

actuating simultaneously at different scales, was handled in

the present work by the use of a methodology that integrated

the results at the micro and meso scales directly with the

macroscopic balances for the reactor. The multigrain model

was used at the meso scale together with comprehensive

local balances for the reactive species and for the moments,

while the kinetic model included the existence of two types

of catalytic sites. The dynamic model of the reactor

accounts for the effects of the different inputs.

The simulation of the reactor model made use of the

orthogonal collocation technique to solve the multigrain

model. A small number of interior collocation points was

enough to provide a satisfactory description of the process

dynamics, thus leading to an expressive reduction of the

computational effort.

The results are qualitatively consistent with the expected

behavior for the process variables and polymer properties.

The hydrogen/ethylene ratio predictions follow the domi-

nant tendency of plant data, even without the existence of

measurements of hydrogen feed flow that has a pronounced

effect in this case.

Acknowledgements

The authors acknowledge the financial support provided

by CNPq through a grant to C. Fontes as well as the

technical support from Polialden S.A.

Appendix A. Appendix

Let �Gn;m be the mean concentration (mol/unit volume

catalyst) of an entity associated exclusively to the macro-

particle, like, for example, Pj(0,0), P1j(n,m), ln;mj . The

macroscopic balance for such entity is:

d½ �Gn;mðtÞVcðtÞ�

dt

Z �Gon;mðtÞ _Vce C �RGn;m

ðtÞVcðtÞK �Gn;mðtÞ _Vcs (A1)

where Vc is the volume of catalyst in reactor, _Vce and _Vcs are

respectively the flow rates of catalyst entering and leaving

the reactor, �RGn;mis the mean rate of production of Gn,m (per

unit volume of catalyst), and Gon;m is the value of Gn,m in the

catalyst entering the reactor.

With the macroscopic balance for the catalyst

dVcðtÞZ _Vce K _Vcs;

dt(A2)

becomes

d �Gn;mðtÞ

dtZ �RGn;m

ðtÞC ð �Gon;mðtÞK �Gn;mðtÞÞ

_Vce

VcðtÞ(A3)

The mean variables �RGn;mand �Gn;m are related to its local

values inside the macroparticles by

�RGn;mZ

ÐVm

RGn;mðr; tÞ½ð1K3Þf3�dVmÐ

Vm½ð1K3Þf3�dVm

(A4)

and

�Gn;mðtÞZ

ÐVm

Gn;mðr; tÞ½ð1K3Þf3�dVmÐVm½ð1K3Þf3�dVm

(A5)

where

RGðr; tÞX2

jZ1

RiGðr; tÞ; Gðr; tÞ

X2

jZ1

Gjðr; tÞ:

But,ðVo

m

½ð1K3Þ=f3�dVom Z

ðVm

½ð1K3Þ=f3�dVm: (A6)

Because the volume of catalyst in the macroparticle is

constant. As there is no flow of Gn,m through the surface of

the macroparticle, and the initial distribution Gon;m may be

considered uniform, one obtains, from equations (A3)–(A5):ðVm

vGn;mðr; tÞ

vt½ð1K3Þ=f3�dVm

Z

ðVm

RGn;mðr; tÞ½ð1K3Þ=f3�dV

C_Vce

VcðtÞ

ðVm

Gon;m½ð1K3Þ=f3�dVm

264

K

ðVm

Gn;mðr; tÞ½ð1K3Þ=f3�dVm

375;


or, finally

vGn;mðr; tÞ

vtZRGn;m

ðr; tÞC Gon;m KGn;mðr; tÞ

� � _Vce

VcðtÞ(A7)

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analysis of an industrial continuous slurry reactor for ethylene–butene copolymerization

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